Given-that-the-sum-of-infinity-of-geometric-series-a-2ar-4ar-2-8ar-3-a-2r-n-1-is-3-and-the-sum-of-infinity-of-geometric-series-a-ar-ar-2-ar-3-ar-n-1-is-k-find-the-range-of-

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Question Number 115946 by ZiYangLee last updated on 29/Sep/20

$$\mathrm{Given}\mathrm{that}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{infinity}\mathrm{of}\mathrm{geometric}$$$$\mathrm{series}a-2ar+4{ar}^2-8{ar}^3+\dots a{(-2r)}^{n-1}+\dots \mathrm{is}3$$$$\mathrm{and}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{infinity}\mathrm{of}\mathrm{geometric}\mathrm{series}$$$$a+ar+{ar}^2+{ar}^3+\dots {ar}^{n-1}+\dots \mathrm{is}k,$$$$\mathrm{find}\mathrm{the}\mathrm{range}\mathrm{of}\mathrm{values}\mathrm{of}k.$$

Answered by Dwaipayan Shikari last updated on 29/Sep/20

$$\mathrm{a}-2\mathrm{ar}+{4\mathrm{ar}}^2-{8\mathrm{ar}}^3+\dots .=\mathrm{a}\left(\frac{1}{1+2\mathrm{r}}\right)$$$$\frac{\mathrm{a}}{1+2\mathrm{r}}=3\Rightarrow 3+6\mathrm{r}=\mathrm{a}\Rightarrow \mathrm{r}=\frac{\mathrm{a}-3}{6}$$$$\mathrm{a}(1+\mathrm{r}+{\mathrm{r}}^2+\dots )=\frac{\mathrm{a}}{1-\mathrm{r}}=\frac{\mathrm{a}}{1-\frac{\mathrm{a}-3}{6}}=\frac{\mathrm{a}}{9-\mathrm{a}}=\mathrm{k}$$$$\mathrm{k}\{0,\frac{1}{8},\frac{2}{7},\frac{1}{2},\frac{4}{5},\frac{5}{4},2,\frac{7}{2},8\}$$$$0⩽\mathrm{k}⩽8$$$$$$

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